1 Introduction
In forming our queuing model for SGH PAC, the role and aim in this section is to extract distributions for these two variables:
- Queueing time. This is the variable that indicates the amount of time taken between the arrival of a patient and the consult with the anaesthesiologist.
- Service time. This is the variable that indicates the amount of time taken in a consult with the anaesthesiologist.
The data given to us is from 2019 to 2021 for both planned appointments and walk-in appointments. However, we will only focus on data from 2020 onwards due to SGH’s claim that the 2019 data may not be as reliable.
2 Assumptions
Acknowledging that the real world data may not be ideal, we would like to lay down some assumptions regarding the data that will help with the model-fitting.
- There is no exact data for queueing time. Therefore, it will be assumed that the queueing time is the total duration that the patient is in PAC subtracted by the actual consult duration. Implicitly, the other PAC stations (e.g. Radiology) where patients are assessed are considered as a “queue” in this case. For our purposes, this decision is justified as the project focuses on the anaesthesiologists and the consults thereof.
- That any service time below 15 minutes is considered a data error. We must also acknowledge the assumption that any data above that is fully accurate to a reasonable degree, except when in the case that it is unrealistically high.
- Seasonality near the Chinese New Year, which is usually in February. This information is provided to us by SGH through their past experience.
3 Methodology
This is a mostly quantitative analysis. The data will be cleaned, meaning that any data error will be removed, and appropriate data types shall be applied to each variable. Data from 2020 and 2021 will then be aggregated into two tibbles (in essence, a better version of a data frame for R), one for planned appointments and walk-in appointments. From the master tibble, they will be split based on both seasonality (February versus the rest of the months) and ASA Score (measure of health in a pre-surgery assessment). There will be sixteen tibbles formed through this split, knowing that ASA Scores range from 1 to 4. However, there may not be enough data to form a distribution for the ASA Score of 4 due to its rarity.
The sixteen tibbles will be analysed using descriptive statistics in the beginning, and then we will try to fit likely distributions to it. Ideally, the distributions will all be exponential. The main test that will be done to corroborate the distribution and its parameters will be the chi-squared test at \(\alpha\) = 0.05.
4 Data Cleaning
4.1 Raw Data
4.1.1 2020 Planned Appointments
## # A tibble: 6 x 21
## `PAC Date` `Visit Time` Identifier Age `Visit dept. OU` `Procedure Code`
## <chr> <chr> <chr> <chr> <chr> <chr>
## 1 1/7/20 8:00 44013879ZGG 63 H&N - ENT SJ802T
## 2 1/7/20 8:00 44013201IGJ 71 Orthop Surg SB741S
## 3 1/7/20 8:00 44013138HIG 36 Orthop Surg SB700H
## 4 1/7/20 8:00 44013339ZGG 47 Gynaecology SI710V
## 5 1/7/20 8:30 44013492ZAZ 3 Otolary SM701E
## 6 1/7/20 8:30 44013516ZGH 69 Orthop Surg SB810K
## # … with 15 more variables: No Show / Attended <chr>, Listing Date <chr>,
## # No. of days between PAC date and listing date <chr>, Surgery Date <chr>,
## # Lead time to Surgery <chr>, Admit Specialty <chr>, ASA Score <chr>,
## # PAC Registration Time <chr>, PAC End Time <chr>,
## # PAC Consult Duration <chr>, TOSP Table Code <chr>,
## # Complexity of Surgery <chr>, TPS Patient (Yes/No) <chr>,
## # TPS Patient (Recruited before PAC / Onsite) <chr>, X21 <lgl>
4.1.2 2020 Walk-In Appointments
## # A tibble: 6 x 20
## `PAC Date` `Visit Time` Identifier Age `Visit dept. OU` `Procedure Code`
## <chr> <chr> <chr> <chr> <chr> <chr>
## 1 1/7/20 8:00 44013585ENB 22 Orthop Surg SB703K
## 2 1/7/20 10:00 44013980AKI 35 Orthop Surg SB840H
## 3 1/7/20 11:00 44013923ABC 66 VASCULAR SUR SD817A
## 4 1/7/20 9:30 44013272ANF 67 Colorec Surg SF802C
## 5 1/7/20 10:00 44013680FGG 85 Colorec Surg SF802C
## 6 1/7/20 14:30 44013335III 47 Gynaecology SI725U
## # … with 14 more variables: No Show / Attended <chr>, Listing Date <chr>,
## # No. of days between PAC date and listing date <chr>, Surgery Date <chr>,
## # Lead time to Surgery <chr>, Admit Specialty <chr>, ASA Score <chr>,
## # PAC Registration Time <chr>, PAC End Time <chr>,
## # PAC Consult Duration <chr>, TOSP Table Code <chr>,
## # Complexity of Surgery <chr>, TPS Patient (Yes/No) <chr>,
## # TPS Patient (Recruited before PAC / Onsite) <chr>
4.1.3 2021 Planned Appointments
## # A tibble: 6 x 20
## `PAC Date` `Visit Time` Identifier Age `Visit dept. OU` `Procedure Code`
## <chr> <time> <chr> <chr> <chr> <chr>
## 1 4/1/21 08:00 44200586JIG 23 Urology SJ802T
## 2 4/1/21 08:00 44200015FDG 57 GI/BARIATRIC SB715K
## 3 4/1/21 08:00 44200377JGH 88 Orthop Surg SB816S
## 4 4/1/21 08:00 44200448ZNC 75 GI/BARIATRIC SB727S
## 5 4/1/21 08:00 44200247HKC #N/A EYE #N/A
## 6 4/1/21 08:00 44200829FEF 74 HRM SB810K
## # … with 14 more variables: No Show / Attended <chr>, Listing Date <chr>,
## # No. of days between PAC date and listing date <chr>, Surgery Date <chr>,
## # Lead time to Surgery <chr>, Admit Specialty <chr>, ASA Score <chr>,
## # PAC Registration Time <chr>, PAC End Time <chr>,
## # PAC Consult Duration <chr>, TOSP Table Code <chr>,
## # Complexity of Surgery <chr>, TPS Patient (Yes/No) <chr>,
## # TPS Patient (Recruited before PAC / Onsite) <chr>
4.1.4 2021 Walk-In Appointments
## # A tibble: 6 x 21
## `PAC Date` `Visit Time` Identifier Age `Visit dept. OU` `Procedure Code`
## <chr> <chr> <chr> <chr> <chr> <chr>
## 1 4/1/21 13:30 44200185FGB 62 Colorec Surg SF836A
## 2 4/1/21 15:00 44200222HGZ 37 Gynaecology SI843U
## 3 4/1/21 8:00 44200230EID 27 Gynaecology SF836A
## 4 5/1/21 14:30 44201892EEG 41 GI/BARIATRIC SF703S
## 5 5/1/21 11:30 44201820DOH 85 VASCULAR SUR SD712A
## 6 5/1/21 13:30 44201147ANI 76 VASCULAR SUR SD821A
## # … with 15 more variables: No Show / Attended <chr>, Listing Date <chr>,
## # No. of days between PAC date and listing date <chr>, Surgery Date <chr>,
## # Lead time to Surgery <chr>, Admit Specialty <chr>, ASA Score <chr>,
## # PAC Registration Time <chr>, PAC End Time <chr>,
## # PAC Consult Duration <chr>, TOSP Table Code <chr>,
## # Complexity of Surgery <chr>, TPS Patient (Yes/No) <chr>,
## # TPS Patient (Recruited before PAC / Onsite) <chr>, X21 <lgl>
4.2 Cleaned Data
4.2.1 Planned Appointments Aggregate
## # A tibble: 6 x 3
## asa queue_time consult_duration
## <chr> <int> <int>
## 1 1 52 33
## 2 2 69 46
## 3 1 72 28
## 4 1 28 20
## 5 2 90 30
## 6 2 413 17
| ASA Score | Queue Time | Consult Duration | |
|---|---|---|---|
| Min. :1.000 | Min. : 13.0 | Min. : 15.00 | |
| 1st Qu.:2.000 | 1st Qu.: 94.0 | 1st Qu.: 21.00 | |
| Median :2.000 | Median :131.0 | Median : 29.00 | |
| Mean :1.822 | Mean :144.6 | Mean : 35.18 | |
| 3rd Qu.:2.000 | 3rd Qu.:180.0 | 3rd Qu.: 42.00 | |
| Max. :4.000 | Max. :556.0 | Max. :319.00 |
Histogram for queuing time for appointments in PAC
Histogram for consult duration for appointments in PAC
Seeing the histogram for consult duration, it might be a bit more useful to think of it as the amount of time beyond 15 minutes the consult will take. Hence, we will translate the consult duration by 15 minutes to the left.
Histogram for adjusted consult duration for appointments in PAC
4.2.2 Walk-In Appointments Aggregate
## # A tibble: 6 x 3
## asa queue_time consult_duration
## <chr> <dbl> <int>
## 1 1 80 17
## 2 3 189 41
## 3 3 136 74
## 4 2 334 23
## 5 2 291 68
## 6 1 59 18
| ASA Score | Queue Time | Consult Duration | |
|---|---|---|---|
| Min. :1.000 | Min. : 16.0 | Min. : 15.00 | |
| 1st Qu.:1.000 | 1st Qu.: 95.0 | 1st Qu.: 20.00 | |
| Median :2.000 | Median :139.0 | Median : 28.00 | |
| Mean :1.794 | Mean :160.9 | Mean : 34.61 | |
| 3rd Qu.:2.000 | 3rd Qu.:206.0 | 3rd Qu.: 40.00 | |
| Max. :3.000 | Max. :585.0 | Max. :185.00 |
Similarly to the planned appointments, the three figures below describe the queue time and consult duration in histograms.
Histogram for queuing time for walk-ins in PAC
Histogram for consult duration for walk-ins in PAC
Histogram for adjusted consult duration for walk-ins in PAC
4.2.3 Appointments Aggregate
## # A tibble: 6 x 3
## asa queue_time consult_duration
## <chr> <dbl> <int>
## 1 1 52 33
## 2 2 69 46
## 3 1 72 28
## 4 1 28 20
## 5 2 90 30
## 6 2 413 17
Histogram for queuing time in PAC
Histogram for consult duration in PAC
Histogram for adjusted consult duration in PAC
5 Finding Distributions
5.1 Planned and Walk-In Appointments
5.1.1 Queue Time
5.1.1.1 ASA 1
## summary statistics
## ------
## min: 13 max: 585
## median: 115
## mean: 129.4893
## estimated sd: 67.28252
## estimated skewness: 1.78132
## estimated kurtosis: 8.385513
5.1.1.1.1 Testing for Lognormal Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.0179067
## Cramer-von Mises statistic 0.0603894
## Anderson-Darling statistic 0.6085893
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 23851.54
## Bayesian Information Criterion 23862.92
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 4.7441447
## sdlog 0.4887809
## Loglikelihood: -11923.77 AIC: 23851.54 BIC: 23862.92
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 3.790103 | 9.487729 |
5.1.1.1.2 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.05277488
## Cramer-von Mises statistic 1.55534146
## Anderson-Darling statistic 12.68223134
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion 23981.55
## Bayesian Information Criterion 23992.94
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 3.70562271
## rate 0.02861722
## Loglikelihood: -11988.78 AIC: 23981.55 BIC: 23992.94
| Pass | Error | Critical Value |
|---|---|---|
| Rejected | 39.74268 | 9.487729 |
5.1.1.2 ASA 2
## summary statistics
## ------
## min: 27 max: 569
## median: 137
## mean: 150.9156
## estimated sd: 71.03786
## estimated skewness: 1.191075
## estimated kurtosis: 4.97947
5.1.1.2.1 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.02898534
## Cramer-von Mises statistic 1.13535433
## Anderson-Darling statistic 10.55945776
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion 69662.87
## Bayesian Information Criterion 69676.35
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 4.51396193
## rate 0.02991051
## Loglikelihood: -34829.43 AIC: 69662.87 BIC: 69676.35
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 17.96091 | 18.30704 |
5.1.1.2.2 Testing for Lognormal Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.02405473
## Cramer-von Mises statistic 0.69879880
## Anderson-Darling statistic 4.89547405
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 69527.77
## Bayesian Information Criterion 69541.25
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 4.9166666
## sdlog 0.4473345
## Loglikelihood: -34761.88 AIC: 69527.77 BIC: 69541.25
| Pass | Error | Critical Value |
|---|---|---|
| Rejected | 29.33012 | 19.67514 |
5.1.1.3 ASA 3
## summary statistics
## ------
## min: 47 max: 495
## median: 137
## mean: 154.6518
## estimated sd: 73.04713
## estimated skewness: 1.272831
## estimated kurtosis: 5.050589
5.1.1.3.1 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.05181065
## Cramer-von Mises statistic 0.35534652
## Anderson-Darling statistic 2.48916670
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion 6256.461
## Bayesian Information Criterion 6265.117
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 4.4903452
## rate 0.0290352
## Loglikelihood: -3126.23 AIC: 6256.461 BIC: 6265.117
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 13.37659 | 18.30704 |
5.1.2 Consult Duration
Adjusted version, meaning that the consult duration was shifted by 15 minutes, unless otherwise stated.
5.1.2.1 ASA 1
## summary statistics
## ------
## min: 0 max: 204
## median: 11
## mean: 16.0456
## estimated sd: 18.30492
## estimated skewness: 3.079084
## estimated kurtosis: 18.48056
5.1.2.1.1 Testing for Lognormal Distribution
The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.09808718
## Cramer-von Mises statistic 2.65276066
## Anderson-Darling statistic 38.40126539
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 16941.03
## Bayesian Information Criterion 16952.41
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 2.4525327
## sdlog 0.8756247
## Loglikelihood: -8468.513 AIC: 16941.03 BIC: 16952.41
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 9.605517 | 15.50731 |
5.1.2.1.2 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.07372338
## Cramer-von Mises statistic 2.59119338
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion -Inf
## Bayesian Information Criterion -Inf
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 0.76873108
## rate 0.04790915
## Loglikelihood: Inf AIC: -Inf BIC: -Inf
| Pass | Error | Critical Value |
|---|---|---|
| Rejected | 55.54124 | 14.06714 |
5.1.2.1.3 Testing for Exponential Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-exp
## Kolmogorov-Smirnov statistic 0.05380757
## Cramer-von Mises statistic 0.70027975
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-exp
## Akaike's Information Criterion 16561.06
## Bayesian Information Criterion 16566.75
##
## [[2]]
## Fitting of the distribution ' exp ' by matching moments
## Parameters :
## estimate
## rate 0.06232238
## Loglikelihood: -8279.528 AIC: 16561.06 BIC: 16566.75
| Pass | Error | Critical Value |
|---|---|---|
| Rejected | 42.66968 | 15.50731 |
5.1.2.2 ASA 2
## summary statistics
## ------
## min: 1 max: 305
## median: 16
## mean: 21.73269
## estimated sd: 21.55089
## estimated skewness: 2.857424
## estimated kurtosis: 18.73142
5.1.2.2.1 Testing for Lognormal Distribution
The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.09597839
## Cramer-von Mises statistic 9.78075237
## Anderson-Darling statistic 146.12106517
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 51935.91
## Bayesian Information Criterion 51949.39
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 2.7364665
## sdlog 0.8274675
## Loglikelihood: -25965.95 AIC: 51935.91 BIC: 51949.39
| Pass | Error | Critical Value |
|---|---|---|
| Rejected | 45.55516 | 5.991465 |
5.1.2.2.2 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.05101785
## Cramer-von Mises statistic 2.85874812
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion -Inf
## Bayesian Information Criterion -Inf
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 0.92565787
## rate 0.04464725
## Loglikelihood: Inf AIC: -Inf BIC: -Inf
| Pass | Error | Critical Value |
|---|---|---|
| Rejected | 15.25287 | 3.841459 |
5.1.2.2.3 Testing for Exponential Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-exp
## Kolmogorov-Smirnov statistic 0.03533173
## Cramer-von Mises statistic 1.08195066
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-exp
## Akaike's Information Criterion 50438.72
## Bayesian Information Criterion 50445.46
##
## [[2]]
## Fitting of the distribution ' exp ' by matching moments
## Parameters :
## estimate
## rate 0.048233
## Loglikelihood: -25218.36 AIC: 50438.72 BIC: 50445.46
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 5.7713 | 5.991465 |
5.1.2.3 ASA 3
## summary statistics
## ------
## min: 1 max: 292
## median: 22
## mean: 30.45536
## estimated sd: 30.72181
## estimated skewness: 2.914877
## estimated kurtosis: 17.01842
5.1.2.3.1 Testing for Lognormal Distribution
The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.06304681
## Cramer-von Mises statistic 0.35976253
## Anderson-Darling statistic 6.46302707
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 4986.764
## Bayesian Information Criterion 4995.420
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 3.0657644
## sdlog 0.8372545
## Loglikelihood: -2491.382 AIC: 4986.764 BIC: 4995.42
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 5.50356 | 12.59159 |
5.1.2.3.2 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.08827393
## Cramer-von Mises statistic 0.73718487
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion -Inf
## Bayesian Information Criterion -Inf
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 0.92089709
## rate 0.03126416
## Loglikelihood: Inf AIC: -Inf BIC: -Inf
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 5.842161 | 5.991465 |
5.1.2.3.3 Testing for Exponential Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-exp
## Kolmogorov-Smirnov statistic 0.06998185
## Cramer-von Mises statistic 0.40160340
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-exp
## Akaike's Information Criterion 4910.821
## Bayesian Information Criterion 4915.149
##
## [[2]]
## Fitting of the distribution ' exp ' by matching moments
## Parameters :
## estimate
## rate 0.03394968
## Loglikelihood: -2454.41 AIC: 4910.821 BIC: 4915.149
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 4.483978 | 7.814728 |
5.2 Planned Appointments
5.2.1 Queue Time
5.2.1.1 ASA 1
## summary statistics
## ------
## min: 13 max: 556
## median: 114
## mean: 127.7265
## estimated sd: 64.52942
## estimated skewness: 1.79333
## estimated kurtosis: 8.60614
5.2.1.1.1 Testing for Lognormal Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.01654383
## Cramer-von Mises statistic 0.06195682
## Anderson-Darling statistic 0.63112144
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 21557.21
## Bayesian Information Criterion 21568.41
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 4.7362772
## sdlog 0.4766837
## Loglikelihood: -10776.6 AIC: 21557.21 BIC: 21568.41
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 5.192523 | 9.487729 |
5.2.1.1.2 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.05008021
## Cramer-von Mises statistic 1.31740467
## Anderson-Darling statistic 10.95928126
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion 21678.92
## Bayesian Information Criterion 21690.12
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 3.91979880
## rate 0.03068901
## Loglikelihood: -10837.46 AIC: 21678.92 BIC: 21690.12
| Pass | Error | Critical Value |
|---|---|---|
| Rejected | 22.16605 | 7.814728 |
5.2.1.2 ASA 2
## summary statistics
## ------
## min: 27 max: 535
## median: 137
## mean: 149.8057
## estimated sd: 69.58894
## estimated skewness: 1.183166
## estimated kurtosis: 4.995085
5.2.1.2.1 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.02925145
## Cramer-von Mises statistic 0.92462179
## Anderson-Darling statistic 9.11450436
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion 64053.78
## Bayesian Information Criterion 64067.10
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 4.6350196
## rate 0.0309402
## Loglikelihood: -32024.89 AIC: 64053.78 BIC: 64067.1
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 20.84311 | 21.02607 |
5.2.1.2.2 Testing for Lognormal Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.02588754
## Cramer-von Mises statistic 0.74568786
## Anderson-Darling statistic 5.00402712
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 63930.79
## Bayesian Information Criterion 63944.11
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 4.9116591
## sdlog 0.4419957
## Loglikelihood: -31963.4 AIC: 63930.79 BIC: 63944.11
| Pass | Error | Critical Value |
|---|---|---|
| Rejected | 38.91663 | 22.36203 |
5.2.1.3 ASA 3
## summary statistics
## ------
## min: 47 max: 480
## median: 136
## mean: 151.883
## estimated sd: 69.55987
## estimated skewness: 1.204725
## estimated kurtosis: 4.808147
5.2.1.3.1 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.0467720
## Cramer-von Mises statistic 0.2687312
## Anderson-Darling statistic 1.8789186
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion 5691.367
## Bayesian Information Criterion 5699.847
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 4.77692467
## rate 0.03145134
## Loglikelihood: -2843.683 AIC: 5691.367 BIC: 5699.847
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 13.61228 | 14.06714 |
5.2.2 Consult Duration
Adjusted version, meaning that the consult duration was shifted by 15 minutes, unless otherwise stated.
5.2.2.1 ASA 1
## summary statistics
## ------
## min: 0 max: 204
## median: 11
## mean: 16.10571
## estimated sd: 18.33906
## estimated skewness: 3.129485
## estimated kurtosis: 19.16101
5.2.2.1.1 Testing for Lognormal Distribution
The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.0954679
## Cramer-von Mises statistic 2.2509975
## Anderson-Darling statistic 34.5996657
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 15441.94
## Bayesian Information Criterion 15453.14
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 2.456952
## sdlog 0.874598
## Loglikelihood: -7718.972 AIC: 15441.94 BIC: 15453.14
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 16.66601 | 16.91898 |
5.2.2.1.2
5.2.2.1.3 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.07696044
## Cramer-von Mises statistic 2.50299959
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion -Inf
## Bayesian Information Criterion -Inf
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 0.7716552
## rate 0.0479119
## Loglikelihood: Inf AIC: -Inf BIC: -Inf
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 13.26114 | 14.06714 |
5.2.2.1.4 Testing for Exponential Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-exp
## Kolmogorov-Smirnov statistic 0.05511022
## Cramer-von Mises statistic 0.63235091
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-exp
## Akaike's Information Criterion 15088.46
## Bayesian Information Criterion 15094.06
##
## [[2]]
## Fitting of the distribution ' exp ' by matching moments
## Parameters :
## estimate
## rate 0.06208978
## Loglikelihood: -7543.231 AIC: 15088.46 BIC: 15094.06
| Pass | Error | Critical Value |
|---|---|---|
| Rejected | 34.53983 | 16.91898 |
5.2.2.2 ASA 2
## summary statistics
## ------
## min: 1 max: 305
## median: 16
## mean: 21.73328
## estimated sd: 21.38637
## estimated skewness: 2.89165
## estimated kurtosis: 19.4802
5.2.2.2.1 Testing for Lognormal Distribution
The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.09452814
## Cramer-von Mises statistic 8.75709084
## Anderson-Darling statistic 133.97976693
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 47926.62
## Bayesian Information Criterion 47939.94
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 2.7402942
## sdlog 0.8228612
## Loglikelihood: -23961.31 AIC: 47926.62 BIC: 47939.94
| Pass | Error | Critical Value |
|---|---|---|
| Rejected | 21.42409 | 7.814728 |
5.2.2.2.2 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.0522794
## Cramer-von Mises statistic 2.6976054
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion -Inf
## Bayesian Information Criterion -Inf
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 0.9400198
## rate 0.0453387
## Loglikelihood: Inf AIC: -Inf BIC: -Inf
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 5.40839 | 5.991465 |
5.2.2.2.3 Testing for Exponential Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-exp
## Kolmogorov-Smirnov statistic 0.03887442
## Cramer-von Mises statistic 1.26676599
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-exp
## Akaike's Information Criterion 46528.28
## Bayesian Information Criterion 46534.94
##
## [[2]]
## Fitting of the distribution ' exp ' by matching moments
## Parameters :
## estimate
## rate 0.04823165
## Loglikelihood: -23263.14 AIC: 46528.28 BIC: 46534.94
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 11.10927 | 12.59159 |
5.2.2.3 ASA 3
## summary statistics
## ------
## min: 1 max: 292
## median: 22
## mean: 30.81287
## estimated sd: 31.43551
## estimated skewness: 2.907392
## estimated kurtosis: 16.71943
5.2.2.3.1 Testing for Lognormal Distribution
The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.0636774
## Cramer-von Mises statistic 0.3279006
## Anderson-Darling statistic 5.7790015
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 4577.690
## Bayesian Information Criterion 4586.171
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 3.0717530
## sdlog 0.8440134
## Loglikelihood: -2286.845 AIC: 4577.69 BIC: 4586.171
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 4.778335 | 7.814728 |
5.2.2.3.2 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.09086619
## Cramer-von Mises statistic 0.72208232
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion -Inf
## Bayesian Information Criterion -Inf
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 0.90118456
## rate 0.03022804
## Loglikelihood: Inf AIC: -Inf BIC: -Inf
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 8.379399 | 9.487729 |
5.2.2.3.3 Testing for Exponential Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-exp
## Kolmogorov-Smirnov statistic 0.06771132
## Cramer-von Mises statistic 0.34491026
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-exp
## Akaike's Information Criterion 4511.208
## Bayesian Information Criterion 4515.449
##
## [[2]]
## Fitting of the distribution ' exp ' by matching moments
## Parameters :
## estimate
## rate 0.03354257
## Loglikelihood: -2254.604 AIC: 4511.208 BIC: 4515.449
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 6.057588 | 11.0705 |
5.3 Walk-In Appointments
5.3.1 Queue Time
5.3.1.1 ASA 1
## summary statistics
## ------
## min: 16 max: 585
## median: 127
## mean: 147.3503
## estimated sd: 88.83391
## estimated skewness: 1.380647
## estimated kurtosis: 5.790429
5.3.1.1.1 Testing for Lognormal Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.0596964
## Cramer-von Mises statistic 0.1086712
## Anderson-Darling statistic 0.7974009
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 2269.386
## Bayesian Information Criterion 2275.953
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 4.8384769
## sdlog 0.5555817
## Loglikelihood: -1132.693 AIC: 2269.386 BIC: 2275.953
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 5.394277 | 9.487729 |
5.3.1.1.2 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.04825587
## Cramer-von Mises statistic 0.08126044
## Anderson-Darling statistic 0.60684980
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion 2265.978
## Bayesian Information Criterion 2272.544
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 2.76537723
## rate 0.01876737
## Loglikelihood: -1130.989 AIC: 2265.978 BIC: 2272.544
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 2.535864 | 7.814728 |
5.3.1.2 ASA 2
## summary statistics
## ------
## min: 28 max: 569
## median: 143
## mean: 164.1196
## estimated sd: 85.41705
## estimated skewness: 1.072297
## estimated kurtosis: 4.166928
5.3.1.2.1 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.04899709
## Cramer-von Mises statistic 0.20490028
## Anderson-Darling statistic 1.22771882
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion 5576.171
## Bayesian Information Criterion 5584.539
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 3.69937258
## rate 0.02254071
## Loglikelihood: -2786.086 AIC: 5576.171 BIC: 5584.539
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 21.08321 | 22.36203 |
5.3.1.2.2 Testing for Lognormal Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.03889669
## Cramer-von Mises statistic 0.18078797
## Anderson-Darling statistic 1.48333130
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 5577.235
## Bayesian Information Criterion 5585.603
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 4.980962
## sdlog 0.489148
## Loglikelihood: -2786.618 AIC: 5577.235 BIC: 5585.603
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 18.26444 | 22.36203 |
5.3.1.3 ASA 3
## summary statistics
## ------
## min: 49 max: 495
## median: 173
## mean: 184.8723
## estimated sd: 99.8424
## estimated skewness: 1.078161
## estimated kurtosis: 3.911473
5.3.1.3.1 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.11069041
## Cramer-von Mises statistic 0.07231369
## Anderson-Darling statistic 0.42905363
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion 556.3347
## Bayesian Information Criterion 560.0350
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 3.50311077
## rate 0.01894881
## Loglikelihood: -276.1673 AIC: 556.3347 BIC: 560.035
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 7.495405 | 14.06714 |
5.3.2 Consult Duration
Adjusted version, meaning that the consult duration was shifted by 15 minutes, unless otherwise stated.
5.3.2.1 ASA 1
## summary statistics
## ------
## min: 0 max: 103
## median: 10
## mean: 15.43655
## estimated sd: 17.98986
## estimated skewness: 2.553924
## estimated kurtosis: 11.12355
5.3.2.1.1 Testing for Lognormal Distribution
The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.1281475
## Cramer-von Mises statistic 0.4791926
## Anderson-Darling statistic 4.4066376
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 1502.302
## Bayesian Information Criterion 1508.868
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 2.4071329
## sdlog 0.8858606
## Loglikelihood: -749.1508 AIC: 1502.302 BIC: 1508.868
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 0.8173222 | 7.814728 |
5.3.2.1.2 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.09339454
## Cramer-von Mises statistic 0.16358727
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion -Inf
## Bayesian Information Criterion -Inf
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 0.74003981
## rate 0.04794076
## Loglikelihood: Inf AIC: -Inf BIC: -Inf
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 12.19202 | 15.50731 |
5.3.2.1.3 Testing for Exponential Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-exp
## Kolmogorov-Smirnov statistic 0.0764491
## Cramer-von Mises statistic 0.1701794
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-exp
## Akaike's Information Criterion 1474.275
## Bayesian Information Criterion 1477.558
##
## [[2]]
## Fitting of the distribution ' exp ' by matching moments
## Parameters :
## estimate
## rate 0.06478132
## Loglikelihood: -736.1374 AIC: 1474.275 BIC: 1477.558
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 10.57594 | 11.0705 |
5.3.2.2 ASA 2
## summary statistics
## ------
## min: 1 max: 171
## median: 14
## mean: 21.72577
## estimated sd: 23.44355
## estimated skewness: 2.526401
## estimated kurtosis: 12.16981
5.3.2.2.1 Testing for Lognormal Distribution
The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.1140782
## Cramer-von Mises statistic 1.1855089
## Anderson-Darling statistic 13.0342018
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 4010.304
## Bayesian Information Criterion 4018.673
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 2.692986
## sdlog 0.878081
## Loglikelihood: -2003.152 AIC: 4010.304 BIC: 4018.673
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 11.18576 | 12.59159 |
5.3.2.2.2 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.06179999
## Cramer-von Mises statistic 0.21824158
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion -Inf
## Bayesian Information Criterion -Inf
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 0.78319684
## rate 0.03778855
## Loglikelihood: Inf AIC: -Inf BIC: -Inf
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 4.265961 | 9.487729 |
5.3.2.2.3 Testing for Exponential Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-exp
## Kolmogorov-Smirnov statistic 0.05585831
## Cramer-von Mises statistic 0.27792163
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-exp
## Akaike's Information Criterion 3912.437
## Bayesian Information Criterion 3916.621
##
## [[2]]
## Fitting of the distribution ' exp ' by matching moments
## Parameters :
## estimate
## rate 0.0482491
## Loglikelihood: -1955.218 AIC: 3912.437 BIC: 3916.621
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 4.463257 | 12.59159 |
5.3.2.3 ASA 3
## summary statistics
## ------
## min: 1 max: 113
## median: 21
## mean: 26.55319
## estimated sd: 21.2979
## estimated skewness: 1.694304
## estimated kurtosis: 7.55201
5.3.2.3.1 Testing for Lognormal Distribution
The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.
## [[1]]
## Goodness-of-fit statistics
## 1-mme-lnorm
## Kolmogorov-Smirnov statistic 0.1106431
## Cramer-von Mises statistic 0.1018993
## Anderson-Darling statistic 1.5959393
##
## Goodness-of-fit criteria
## 1-mme-lnorm
## Akaike's Information Criterion 419.0401
## Bayesian Information Criterion 422.7404
##
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments
## Parameters :
## estimate
## meanlog 3.0349670
## sdlog 0.6988318
## Loglikelihood: -207.52 AIC: 419.0401 BIC: 422.7404
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 2.577932 | 7.814728 |
5.3.2.3.2 Testing for Gamma Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-gamma
## Kolmogorov-Smirnov statistic 0.05606922
## Cramer-von Mises statistic 0.01871981
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-gamma
## Akaike's Information Criterion Inf
## Bayesian Information Criterion Inf
##
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments
## Parameters :
## estimate
## shape 1.47081119
## rate 0.05755881
## Loglikelihood: -Inf AIC: Inf BIC: Inf
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 2.209917 | 9.487729 |
5.3.2.3.3 Testing for Exponential Distribution
## [[1]]
## Goodness-of-fit statistics
## 1-mme-exp
## Kolmogorov-Smirnov statistic 0.1239516
## Cramer-von Mises statistic 0.1281206
## Anderson-Darling statistic Inf
##
## Goodness-of-fit criteria
## 1-mme-exp
## Akaike's Information Criterion 400.6316
## Bayesian Information Criterion 402.4818
##
## [[2]]
## Fitting of the distribution ' exp ' by matching moments
## Parameters :
## estimate
## rate 0.03913405
## Loglikelihood: -199.3158 AIC: 400.6316 BIC: 402.4818
| Pass | Error | Critical Value |
|---|---|---|
| Not rejected | 6.167518 | 11.0705 |
6 Summary and Conclusion
6.1 Planned and Walk-In Appointments
6.1.1 Queue Time
| ASA Score | Distribution | Error | Critical Values |
|---|---|---|---|
| 1 | Lognormal(4.7441447, 0.4887809) | 3.790103 | 9.487729 |
| 2 | Gamma(4.51396193, 0.02991051) | 17.96091 | 18.30704 |
| 3 | Gamma(4.4903452, 0.0290352) | 13.37659 | 18.30704 |
6.1.2 Consult Duration
| ASA Score | Distribution | Error | Critical Values |
|---|---|---|---|
| 1 | Lognormal(2.4525327, 0.8756247) | 9.605517 | 15.50731 |
| 2 | Exp(0.048233) | 5.7713 | 5.991465 |
| 3 | Lognormal(3.0657644, 0.8372545) | 5.50356 | 12.59159 |
| Gamma(0.92089709, 0.03126416) | 5.842161 | 5.991465 | |
| Exp(0.03394968) | 4.483978 | 7.814728 |
6.2 Planned Appointments
6.2.1 Queue Time
| ASA Score | Distribution | Error | Critical Values |
|---|---|---|---|
| 1 | Lognormal(4.7362772, 0.4766837) | 5.192523 | 9.487729 |
| 2 | Gamma(4.6350196, 0.0309402) | 20.84311 | 21.02607 |
| 3 | Gamma(4.77692467, 0.03145134) | 13.61228 | 14.06714 |
6.2.2 Consult Duration
| ASA Score | Distribution | Error | Critical Values |
|---|---|---|---|
| 1 | Lognormal(2.456952, 0.874598) | 16.66601 | 16.91898 |
| Gamma(0.7716552, 0.0479119) | 13.26114 | 14.06714 | |
| 2 | Gamma(0.9400198, 0.0453387) | 5.40839 | 5.991465 |
| Exp(0.04823165) | 11.10927 | 12.59159 | |
| 3 | Lognormal(3.0717530, 0.8440134) | 4.778335 | 7.814728 |
| Gamma(0.90118456, 0.03022804) | 8.379399 | 9.487729 | |
| Exp(0.03354257) | 6.057588 | 11.0705 |
6.3 Walk-In Appointments
6.3.1 Queue Time
| ASA Score | Distribution | Error | Critical Values |
|---|---|---|---|
| 1 | Lognormal(4.8384769, 0.5555817) | 5.394277 | 9.487729 |
| Gamma(2.76537723, 0.01876737) | 2.535864 | 7.814728 | |
| 2 | Gamma(3.69937258, 0.02254071) | 21.08321 | 22.36203 |
| Lognormal(4.980962, 0.489148) | 18.26444 | 22.36203 | |
| 3 | Gamma(3.50311077, 0.01894881) | 7.495405 | 14.06714 |
6.3.2 Consult Duration
| ASA Score | Distribution | Error | Critical Values |
|---|---|---|---|
| 1 | Lognormal(2.4071329, 0.8858606) | 0.8173222 | 7.814728 |
| Gamma(0.74003981, 0.04794076) | 12.19202 | 15.50731 | |
| Exp(0.06478132) | 10.57594 | 11.0705 | |
| 2 | Lognormal(2.692986, 0.878081) | 11.18576 | 12.59159 |
| Gamma(0.78319684, 0.03778855) | 4.265961 | 9.487729 | |
| Exp(0.0482491) | 4.463257 | 12.59159 | |
| 3 | Lognormal(3.0349670, 0.6988318) | 2.577932 | 7.814728 |
| Gamma(1.47081119, 0.05755881) | 2.209917 | 9.487729 | |
| Exp(0.03913405) | 6.167518 | 11.0705 |